Buoyancy of Gas-filled Envelope

Buoyancy of Gas-filled Envelope

The maximum change of height within a balloon or a gas-bag of an
airship is usually sufficiently small for variation of density to be
neglected. Draw a vertical cylinder of small cross-sectional area A
completely through the envelope E (Fig. 5), which is filled with a
light gas of density p', and is at rest relative to the surrounding
atmosphere of density p. Let the cylinder cut the envelope at a
lower altitude-level ht and at an upper one Aa , the curves of inter-

section enclosing small areas Slt Sa , the normals to which (they are

not necessarily in the same plane) make angles oc lf a8 with the

vertical. On these areas pressures />',, p'2t act outwardly due to the

gas, and^>lf p9 act inwardly due to the atmosphere.

There arises at h2 an upward force on the cylinder equal to

(pi ^a)S2 cos a,.

The similar force arising at h^ may be upward or downward, depending

on the position of Sl and whether an airship or a balloon is

considered, but in any case its upward value is

(Pi P()SI cos i- Since Sa cos oc 8 = A == St cos al, 

the resultant upward force on the

cylinder due to the pressures is

Substituting from (3),